8 problems

Let $T$ be the random tree under consideration, let $t$ denote its size parameter, and let $\operatorname{cov}_\ell(T)$ be the minimum number of paths of length at most $\ell$ need…

Let $G$ be a random $d$-regular bipartite graph with bipartition $V_1\cup V_2$, and for density parameters $(\alpha,\beta)$ let $Z_G^{\alpha,\beta}$ denote the total weight of inde…

Logarithmic maximum-entry conjecture. With high probability,

Let $b0,1d^d$ be the Boolean cube, let $bmu$ be a probability distribution on it with mean $p$, and let $X^{(1)},dots,X^{(n)}$ be an iid sample from $bmu$. Define the empirical mea…

Let $G_{n,p}$ be the binomial random graph with edge-probability $p$, and let $\gamma(G_{n,p})$ denote its domination number, the smallest size of a set of vertices such that every…

Let $M$ be a random $k\times k$ matrix with independent standard complex Gaussian entries, where $k\geq 3$ is an integer, and let $t\geq 3$ be real. The non-integer moment growth c…

Let $u(t,x)$ be a solution whose support has length $\ell(t)=\sigma_+(t)-\sigma_-(t)$, with initial length $\ell(0)=\xi_+-\xi_-$. Suppose that the total mass satisfies…

Typical-cluster entropy concentration conjecture. The random quantity $n^{-1}\log Z_{\ell,n}$ concentrates in probability, as $n\to\infty$, around a non-random value.